Method of generating a control program for a device for photorefractive corneal surgery of the eye

ABSTRACT

Method of generating a control program for a device for photorefractive corneal surgery of the eye In a method of generating a control program, according to which a laser-beam spot is guided, while being controlled with respect to position and time, over a cornea to be corrected photorefractively, so as to ablate a predetermined ablation profile therefrom, the effect of the angle between the laser beam and the corneal surface on the energy density of the laser-beam spot incident on the corneal surface and/or on the fraction of the laser-beam energy incident on the corneal surface which is reflected away, is taken into account when generating the control program.

This application is a divisional of U.S. application Ser. No.10/031,107, filed Jan. 10, 2002; which is a 371 application ofPCT/EP01/04978 filed May 3, 2001; which claims priority to Germanapplication 100 22 995.6, filed May 11, 2000.

The invention relates to a method of generating a control program,according to which a laser-beam spot is guided, while being controlledwith respect to position and time, over a cornea to be corrected, so asto ablate a predetermined ablation profile therefrom. The invention alsorelates to an electronic computer and to a device for corneal surgery ofthe eye, in which a control program generated by means of the method isused.

Photorefractive keratectomy has become a widely established method ofcorrecting lower-order visual disorder, for example myopia, hyperopia,astigmatism, myopic astigmatism and hyperopic astigmatism. The term“photorefractive keratectomy (PRK)” is usually understood as meaningonly intervention on the corneal surface, after the so-called cornealepithelium has been removed. After the epithelium is removed, theBowman's membrane, or corneal stroma, is exposed and can be ablatedusing a laser. Distinction is generally made between PRK and the LASIKmethod (Laser In Situ Keratomileusis). In the LASIK method, a so-calledmicrokeratome is firstly used to excise an approximately 100 μm to 200μm-thick corneal flap with a diameter of from 8 to 10 mm, leaving only asmall remnant which acts as a “hinge”. This flap is folded to the side,and then material is ablated (removed) by means of laser radiationdirectly in the stroma, i.e. not at the corneal surface. After the lasertreatment, the cover is folded back to its original place, andrelatively fast healing generally takes place.

The invention described below is suitable both for the aforementionedPRK and, in particular, for the LASIK technique.

In PRK and in LASIK, material of the cornea is ablated. The ablation isa function of the laser beam's energy density (energy per unit area)incident on the cornea. Various techniques for shaping the beam andguiding the beam are known, for example so-called slit scanning, inwhich the radiation is guided, by means of a moving-slit, over the areato be processed, so-called spot scanning, in which a radiation spot ofvery small dimensions is guided over the region to be ablated, andso-called full ablation, or wide-field ablation, in which the radiationis projected with a wide field over the full area to be ablated, andwhere the energy density changes across the beam profile in order toachieve the desired ablation of the cornea. For the said forms of beamguidance, the prior art contains respectively suitable algorithms forcontrolling the radiation, in order to ablate the cornea in such a waythat the desired radius of curvature is finally imparted to the cornea.

The “spot scanning” already mentioned above uses a laser beam which isfocused onto a relatively small diameter (0.1-2 mm), which is directedat various positions on the cornea by means of a beam-guiding instrumentand which is successively moved, by a so-called scanner, so that thedesired ablation from the cornea is finally achieved. The ablation hencetakes place according to a so-called ablation profile. In particular,so-called galvanometric scanners can be used in PRK and LASIK (cf. thearticle by G. F. Marshall in LASER FOCUS WORLD, June 1994, p. 57). Sincethen, other scan techniques have been disclosed for guiding the laserbeam.

According to the prior art, the said lower-order visual disorders (e.g.myopia, hyperopia, astigmatism) are currently performed [sic] accordingto the so-called refraction data of the patient's eye, i.e. the dioptricvalue measured for the patient's eye dictates the ablation profileaccording to which material will be removed (ablated) from the cornea(cf. T. Seiler and J. Wollensak in LASERS AND LIGHT IN OPHTHALMOLOGY,Vol. 5, No 4, pp. 199-203, 1993). According to this prior art, for agiven patient's eye with a particular dioptric value, the laserradiation is hence guided over the cornea in such a way that apredetermined ablation profile is removed, for example according to aparabola when correcting myopia. In other words: the ablation profile ismatched to the individual eye only according to the dioptric value, butnot according to local non-uniformities of the “eye” optical system.

The article by J. K. Shimmick, W. B. Telfair et al. in JOURNAL OFREFRACTIVE SURGERY, Vol. 13, May/June 1997, pp. 235-245, also describesthe correction of lower-order sight defects by means of photorefractivekeratectomy, where the photoablation profiles correspond to theoreticalparabolic shapes. Furthermore, this citation only proposes that a fewempirical correction factors, which take account of the interactionbetween the laser and the tissue, be added in to the ablation profile inorder thereby to achieve paraboloidal ablation on the eye.

A particular problem in photorefractive keratectomy and LASIK involvesthe relative positioning of the laser beam and the eye. The prior artcontains various methods for this, for example so-called eye-trackers,i.e. instruments which determine the eye's movements so that the laserbeam used for the ablation can then be controlled (tracked) inaccordance with the ocular movements. The prior art relating to this isdescribed, for example, by DE 197 02 335 C1.

As mentioned above, the methods of photorefractive corneal surgery inthe prior art for correcting lower-order visual disorder are essentially“wholesale” methods, in so far as the correction is based on the(wholesale) dioptric value of the eye. Such lower-order visual disordercan be corrected, for example, by spherical or astigmatic lenses, orindeed by photorefractive correction of the cornea.

However, the optical imaging in the eye is impaired not only by the saidlower-order visual disorders, but also by so-called higher-order imagedefects. Such higher-order image defects occur, in particular, afteroperative interventions on the cornea and inside the eye (cataractoperations). Such optical aberrations can be the reason why, despitemedical correction of a lower-order defect, full visual acuity (sight)is not achieved. P. Mierdel,.H.-E. Krink, W. Wigand, M. Kaemmerer and T.Seiler describe, in DER OPHTALMOLOGE [THE OPHTHALMOLOGIST], No 6, 1997,p. 441, a measuring arrangement for identifying the aberration of thehuman eye. With such a measuring arrangement, it is possible to measureaberrations (imaging defects) for monochromatic light, and moreover notonly aberrations due to the cornea, but also the imaging defects causedby the entire ocular imaging system of the eye can be measured, andactually as a function of position, i.e. with a particular resolution itis possible to determine, for given locations inside the pupil of theeye, how great is the imaging defect of the entire optical system of theeye to be corrected, at this position. Such imaging defects of the eyeare mathematically described as a so-called wavefront aberration in thework by P. Mierdel et al. cited above. The term “wavefront aberration”is used to mean the spatial variation of the distance between the actuallight wavefront from a central light point and a reference surface, e.g.its ideal spherical configuration. For instance, the sphere surface ofthe ideal wavefront is used as the spatial reference system. As thereference system for measuring the aberration, a plane is chosen whenthe ideal wavefront to be measured is plane.

The measuring principle according to the said work by P. Mierdel, T.Seiler et al. is also used in PCT/EP00/00827. It essentially involvessplitting a parallel beam bundle of sufficient diameter through a holemask into separate parallel individual beams. These individual beamspass through a converging lens (the so-called aberroscope lens) so thatthey are focused at a particular distance in front of the retina in thecase of an emmetropic'eye. The result is highly visible projections ofthe mask holes on the retina. This retinal light-point pattern isimaged, according to the principle of indirect ophthalmoscopy, onto thesensor surface of a CCD video camera. In the aberration-free ideal eye,the imaged-light-point pattern is undistorted and corresponds exactly tothe hole-mask pattern. If there is an aberration, however, thenindividual displacements of each pattern point occur, because eachindividual beam passes through a particular corneal or pupillary areaand experiences a deviation from the ideal path according to thenon-uniform optical effect. From the retinal pattern-pointdisplacements, the wavefront aberration is finally determined by anapproximation method as a function of position over the pupillarysurface. The said prior art also describes the mathematicalrepresentation of this wavefront aberration in the form of a so-called“wavefront-aberration hill”. Above each pupillary location (x-ycoordinates), this “wavefront-aberration hill” indicates a value of thewavefront aberration W(x,y) which is then plotted as a height above thex-y coordinates. The higher the “hill” is, the greater are the imagingconsumptions [sic] in the eye at the respective pupillary location. Foreach incident light beam, there is to first approximation aproportionality between the measured deviation of the correspondingretinal light point from its ideal position and the gradient of the“wavefront-aberration hill”. The wavefront aberration can hence beidentified from this as a function of position, with respect to anarbitrary reference value on the optical axis of the system. Ideal, ingeneral undistorted light-point positions on the retina, which can yieldthe reference value, are for example four central points at a smalldistance from one another. Such points represent a centralcorneal/pupillary zone of about 1 to 2 mm in diameter, which fromexperience can be assumed to be substantially free of higher-order imagedefects.

The “wavefront-aberration hill” can be mathematically represented invarious ways with the use of a closed expression (a function). Suitableexamples include approximations in the form of a sum of Taylorpolynomials or, in particular, Zernike polynomials. The Zernikepolynomials have the advantage that their coefficients have a directrelationship with the well known image defects (aperture defects, coma,astigmatism, distortion). The Zernike polynomials are a set ofcompletely orthogonal functions. An article by J. Liang, B. Grimm, S.Goelz and J. F. Bille, “Objective Measurement of Wave Aberrations of theHuman Eye with the use of a Hartmann-Shack Wavefront Sensor”, OpticalSociety of America, 11(7): 1949-1957, July 1994, shows how the wavefront(or wavefront aberration) can be calculated from the grid-pointdisplacements. From identifying the derivative function of thewavefront, it is hence possible to determine the actual wavefront. Thewavefront is found as the solution of a system of equations. The articleby H. C. Howland and B. Howland, “A Subjective Method for theMeasurement of Monochromatic Aberrations of the Eye”, Journal of theOptical Society of America 67(11): 1508-1518, November 1977, alsodescribes a method for identifying monochromatic aberration and thedetermination of the first fifteen Taylor coefficients.

The device proposed in the aforementioned PCT/EP00/00827 forphotorefractive corneal surgery in the case of higher-order sightdefects has the following instruments:

-   -   an aberroscope for measuring the wavefront aberration of the        entire optical system of the eye to be corrected, with respect        to a particular ocular position,    -   means for deriving a photoablation profile from the measured        wavefront aberration so that photoablation according to the        photoablation profile minimises the wavefront aberration of the        eye being treated, and    -   a laser-radiation source and means for controlling the laser        radiation with respect to the particular ocular position, in        order to ablate the photoablation profile.

And, if this device produced significant improvements compared with theprevious solutions, it was found that the treatment successes in somecases were not as good as might have been expected in view of theaccuracy with which the photoablation profile had been compiled.

It is therefore an object of the present invention to provide a way inwhich even better treatment successes can be achieved.

The invention is based on the discovery that in the prior art, althougha very accurate ablation profile was determined, the simplifyingassumption was nevertheless made, when carrying out the ablation, thatthe laser beam induces uniform ablation at each location on the cornea.However, the laser beam is incident with different angles at the variouslocations on the cornea. This has two consequences: on the one hand, thedensity of the laser-beam energy incident on the corneal surface changeswith this angle and, on the other hand, a differing fraction of theincident laser radiation is reflected depending on the angle.

The invention correspondingly provides a method of generating a controlprogram of the generic type mentioned in the introduction, in which,when generating the control program, the effect of the angle between thelaser beam and the corneal surface on the energy density of thelaser-beam spot incident on the corneal surface is taken into account.

Alternatively or additionally, account is taken of the fact that afraction of the laser-beam energy incident on the corneal surface isreflected away.

Preferably, in this case, formulae that are described more explicitlybelow are employed.

The invention can even be used when not just one laser-beam spot isapplied to the cornea, but rather a full laser-beam profile, as in thefull ablation mentioned in the introduction and in slit scanning.

The invention furthermore relates to a program medium, as well as to anelectronic computer for delivering control signals to control a laserbeam, the computer being programmed with a control program generatedaccording to the said method, which it runs when delivering the controlsignals.

The invention furthermore relates to a device for photorefractivecorneal surgery of the eye to correct sight defects, having:

-   -   an instrument for measuring the optical system of the eye to be        corrected,    -   means for deriving an ablation profile from the measured values,    -   a laser-radiation source and means for controlling the laser        radiation, the control means comprising an electronic computer        which runs a control program that has been generated using the        method according to the invention.

An exemplary embodiment of the invention will be explained in moredetail below with the aid of the drawings, in which:

FIG. 1 schematically shows the wavefront aberration;

FIG. 2 schematically shows an aberroscope for measuring the wavefrontaberration of the entire optical system of an eye to be treated;

FIG. 3 schematically shows a measuring and control arrangement forcarrying out photorefractive keratectomy of the eye, with means forderiving a photoablation profile and means for controlling the laserbeam;

FIG. 4 shows the dependency of the ablation depth on the beam energydensity;

FIG. 5 schematically shows the surface of the cornea with a laser-beamspot incident on the surface and with axes shown;

FIG. 6 shows the dependency of a first correction factor on the distancer of the incidence point of the laser-beam spot centre on the corneafrom the z axis for different radii R of the cornea;

FIG. 7 schematically shows the surface of the cornea and the laser beamincident at the angle α₁;

FIG. 8 shows the dependency of a second correction factor on thedistance r of the incidence point of the laser-beam spot centre on thecornea from the z axis [lacuna] different radii R of the cornea;

FIG. 9 shows the dependency of a combined correction factor for theablation depth on the distance r of the incidence point of thelaser-beam spot-centre on the cornea from the z axis for different radiiR of the cornea;

FIG. 10 shows the dependency of the ratio of the distance at which thedensity of the unreflected energy incident on the corneal surface is80%, to the distance at which it is 0, on the beam energy density of theincident laser beam;

FIG. 11 schematically shows the beam path for non-centred incidence ofthe laser-beam spot;

FIG. 12 shows the dependency of the combined correction factor for theablation depth on the distance r of the incidence point of thelaser-beam spot centre on the cornea from the z axis shown in FIG. 11with a differing degree of the decentring r_(v) for a radius ofcurvature R=7.8 mm and a beam energy density F=150 mJ/cm² of theincident laser beam.

FIG. 1 schematically shows the aforementioned wavefront aberration of aneye, i.e. the deviation of the real, aspherical wavefront from the idealwavefront. A is the optical axis of the system and F is the focal point,which is also the imaginary starting point of the radiation in the caseof an ideal wavefront.

FIG. 2 schematically shows the optical layout of a video aberroscope formeasuring the wavefront aberration of an eye 10. The green light from aHeNe laser (543 nm) is broadened to a diameter of approximately 12 mmand subsequently split by means of a hole mask 12, in which a pluralityof equidistant holes are formed, into a corresponding number of parallelindividual beams. According to FIG. 2, these individual beams, which areindicated only schematically by dotted lines, run parallel to theoptical axis A of the system. Using an aberroscope lens 14 (converginglens) in front of the eye 10, these beams are refracted in such a waythat they are focused at a particular distance in front of the retina 20(focus F). In the case of an eye with correct version, the aberroscopelens has e.g. a power of +4 dpt. In the aberration-free ideal eye, acompletely undistorted light-point pattern is produced on the retina 20.The pupil is indicated by the reference number 18.

If the eye 10 has an aberration, however, then the pattern points willbe displaced according to the imaging defects, since each individualbeam passes only through a very particular location in the pupil 18 andexperiences a deviation from the ideal path according to the non-uniformoptical effects. This deviation from the ideal path corresponds to theoptical imaging defect of the entire optical system of the eye 10 withthe respect to a light beam which passes through the particular locationinside the pupil. On the cornea, the individual beams have e.g. aconstant separation of 1.0 mm in the x and y directions, and theirdiameter is for example approximately 0.5 mm. The entire parallelmeasurement-beam bundle has e.g. a size of 8×8 mm on the cornea.

By means of a semi-silvered mirror 16, the light-point pattern producedon the retina 20 is imaged via an ophthalmoscope lens 22 and anobjective 24 for the retinal image onto a sensor surface 28 of asolid-state image camera (CCD camera) so that the resulting light-pointpattern can be processed by a computer. The deviations of the locationsof the light points, with respect to the equidistant, uniform structureof the defect-free eye, provides the opportunity to determine thewavefront aberration W(x,y) as a function of position over the pupillarysurface of the eye. The function of position can be approximated byusing a set of polynomials, e.g. Taylor polynomials or Zernikepolynomials. The Zernike polynomials are preferred here because theircoefficients C_(i) have the advantage of a direct relationship with theimage defects, such as aperture defects, coma, astigmatism, distortion.With the Zernike polynomials Z_(i)(x,y), the wavefront aberration W canbe represented as follows:W(x,y)=Σ_(i) C _(i) ×Z _(i)(x,y).

The Cartesian coordinates in the pupillary plane are denoted (x,y).

By identifying e.g. the first 14 coefficients C_(i) (i=1, 2, . . . , 14)of the Zernike polynomials, it is possible to obtain a sufficientlyaccurate description of the wavefront aberration W(x,y) as a function ofthe position coordinates of the free pupillary surface. This provides aso-called wavefront-aberration hill, i.e. in a three-dimensionalrepresentation, a function above the position coordinates x,y whichindicates the local imaging defects in each case. Apart from the Zernikepolynomials, it is also possible to choose other options formathematically describing the wavefront, e.g. Taylor series. The Zernikepolynomials are merely the exemplary embodiment selected here.

From this wavefront'aberration W(x,y), a so-called photbablation profileis calculated by using a computer 48 (FIG. 3). The computer hencefinally determines, from the light-point pattern, the wavefrontaberration in the form of a particular number of Zernike coefficientsand then determines, from the wavefront aberration, a photoablationprofile, i.e. data concerning the depth to which the cornea needs to beremoved (ablated) at the respective pupillary location in order toreduce the wavefront aberration. The ablation profile, i.e. the layerthickness of the material to be removed as a function of position (X-Ycoordinates) can be identified from the wavefront (aberration) invarious ways:

Basically, the ablation profile for an eye to be corrected is calculatedusing a corresponding eye model.

To that end, the wavefront aberration is mathematically projected ontothe corneal surface while taking into account the geometrical propertiesof the eye, e.g. the thickness of the cornea, the distance between theback surface of the cornea and the front surface of the lens, thedistance between the front surface of the lens and the back surface ofthe lens, and the distance between the back surface of the lens and theretina. The refractive indices of the individual optical elements of theeye are furthermore taken into account when calculating the ablationprofile. The wavefront essentially describes the differences in the timeof flight of the light, i.e. the optical path distance. If the opticalpath difference is divided by the refractive index, then the geometricalpath is obtained. From the projection of the wavefront onto the cornea,it is hence possible to derive the associated ablation profile. In theform of an iteration, an ablation depth (in LASIK, correspondingly adepth of the material ablated in the stroma) is mathematically assumedat the given location on the cornea, and the effect which such ablationwould cause on the time-of-flight differences of the beams iscalculated. The aim is to equalise the times of flight of the beams atall locations on the cornea, so that the wavefront aberration willbecome as small as possible. In this, case, it is necessary to take intoaccount that the wavefront can also assume values whose physical meaningimplies addition of tissue (i.e. thickening of the cornea), which is notgenerally possible. The ablation profile must therefore be adaptedaccordingly, i.e. shifted overall so that the desired target profile ofthe cornea will be achieved only by ablation (removal) of tissue.

The wavefront aberration can be calculated not only in the pupillaryplane (entrance pupil), but also directly at the cornea. When thecorresponding refractive indices are taken into account, the actualablation profile for a particular pupillary diameter is hence obtained.

A correction for the wavefront aberration W(x,y) used to determine theablation profile is made so as to take the eye's healing process afterthe operation into account as well. This is because the healing processleads to a change in the optical properties of the eye, and [lacuna]that these changes [sic] ought to be taken into account in the basicwavefront aberration in order to achieve the best results. This is doneas follows:

So-called correction factors (“fudge factors”) A_(i) are introduced intothe equation above, in which the wavefront aberration W(x,y) isrepresented as a sum of Zernike polynomials Z_(i)(x,y):${W\left( {x,y} \right)} = {\sum\limits_{i = 0}^{n}{A_{i} \times C_{i} \times {Z_{i}\left( {x,y} \right)}}}$

In comparison with the formula above, correction factors A_(i), whichempirically take account of the wound-healing process, have respectivelybeen added in the sum of Zernike, coefficients and Zernike polynomials.In other words: the above function W(x,y) describes the wavefront to becorrected at the eye while taking into account postoperative changes ofindividual optical image defects (Z_(i)) due to the wound healing. Inthis case, the Zernike coefficients of zeroth to eighth order, inparticular, are clinically relevant. As already explained above, thepolynomial coefficients Ci describe the degree of the image defect fromthe described measurement.

It has been empirically shown that the clinically relevant value-range[sic] of the correction factors A_(i) lies in the range of from −1000 to0 to +1000. It has also been empirically determined that the clinicalcorrection factors A_(i) assume different values for each coefficientC_(i). A_(i) is hence a function of C_(i). This functional dependencyA_(i)=f_(i)(C_(i)) differs for the individual coefficients C_(i), i.e.the function f_(i) has different forms for the individual coefficientsC_(i).

It has also been shown that the function A_(i)=f_(i)(C_(i)) isfurthermore dependent on the therapeutic laser system respectively used,since the postoperative healing process is itself also dependent on thelaser system respectively used. This means that generally applicable(abstract) data or algorithms cannot in general be provided for theclinical correction factors A_(i), and these correction factors mustinstead be empirically (experimentally) determined clinically for thelaser system respectively used, the aforementioned typical value rangeof −1000 through 0 to +1000 being applicable, in particular, for thelaser system from WaveLight, Erlangen, Germany which was used here.

As stated, when the aforementioned correction factors A_(i)are not used,ablation profiles determined on the basis of the wavefront aberrationcan lead to overrating or underrating of individual image defects owingto the healing of the wound after the refractive intervention, forinstance, in LASIK, the healing of the flap which had been folded back.To correct a coma of Z₇=0.3 μm, for instance, it is necessary to ablatea coma of about Z₇=0.5 μm from the cornea so that a Z₇=0 (here, “Z”stands for the Zernike coefficient as an example) is obtained at the endof the wound healing (e.g. epithelial closure, approximately 7 days).

The correction factors A_(i) determined as indicated above will storedin the computer, and the computer program worked [sic] them(automatically) into the ablation profile that will finally be used.

As an alternative to the aforementioned calculation of the ablationprofile from the wavefront aberration, the ablation profile may also becalculated directly from projection of points onto the cornea and theretina. If a light beam is incident with known angles of incidence andcoordinate points on the cornea and then in the eye, then this lightbeam will be imaged onto the retina according to the optical propertiesof the eye. Since the position of the light beam on the cornea and theangle of incidence of the beam are known, the optical beam path can bereproduced by measuring the position of the light beam on the cornea. Inthis context, if it is found that the position of the light beam on theretina deviates from the target position (the target position meansaberration-free imaging), then the aberration can be determined from thepositional deviation. The light is refracted according to thegeometrical curvature of the surface of the cornea and the furtheraberration defects of the “eye” system. The aforementioned positionaldeviation of the light beam on the retina can be expressed by acorresponding change in the angle of incidence of the light. The angleof incidence of the light is proportional to the derivative function ofthe surface of the cornea. Through an iterative procedure, a(pathological) change in the curvature of the corneal surface can beinferred from the positional shift of the light beam on the retina andthe associated change in the angle of incidence of the light. The changein the curvature of the corneal surface hence describes the derivativefunction of the (desired) ablation profile. If this method is carriedout with a sufficient number of light beams at different points in theeye (e.g. by projecting a grid onto the cornea), then the entirederivative function of the (desired) ablation profile can be identified.The ablation profile can be calculated from this by known mathematicalmethods (e.g. spline interpolation and subsequent integration).

It has been shown that in some cases, ablation profiles obtained bywavefront measurements make it necessary to have a so-called transitionzone because, in the absence of such a transition zone, a certainresidue of material could possibly be left at the edge of the ablationprofile,. i.e. a step would be created on the cornea. In order to avoidsuch a step, an approximately 0.5 mm to 3 mm-wide transition zone isprovided around the outside of the ablation profile, in order toguarantee a smooth, step-free surface on the entire cornea.

FIG. 3 schematically shows the computer system and control system forcarrying out photoablation according to the computed photoablationprofile. The photoablation takes place both superficially on the corneaand intra-stromally.

An example of a particularly suitable laser 30 for the photoablation isan excimer laser (193 nm). Also particularly suitable are Er: YAGsolid-state lasers with a wavelength of 2.94 μm and UV solid-statelasers (e.g. Nd: YAG with 213 nm).

The laser radiation is deflected by means of a galvanometric scanner 32,and the deflected laser beam 34 is directed at the eye 10.

Coaxially with the laser beam 34, another beam from a so-calledpositioning light source 36 is directed at the eye 10. The beam 50 fromthe positioning light source 36 defines a reference axis A, whoseposition is fixed in space.

In the real case, the eye 10 moves with respect to the axis A. In order,in the event of such movements, to match (track) the processing beam 34,and hence the ablation profile to be removed, to the movements of theeye, the eye is illuminated with infrared radiation (not shown), and theCCD camera 28 is used to record images at a particular image repetitionfrequency. The image radiation 42 from the eye hence generates images inthe CCD camera 28, which are electronically processed. The electronicoutput signal 44 from the camera 28 is sent to an image-processinginstrument 40, and the result of the image processing is input into acomputer 48, which both performs the evaluation and controls the scanner32. The computer 48 hence outputs a corresponding control signal 46 tothe scanner 32, so that the laser beam 34 is controlled in such a waythat the ablation profile too is removed with respect to a particularocular position, with respect to which the wavefront ablation [sic] wasalso measured. In this way, the optical defects of the entire eye can becorrected by photoablation of the cornea. The ablation profile removedhere in the above sense is the ablation profile that was obtained fromthe wavefront measurement and was modified by the aforementionedempirical correction factors because of the wound healing.

The device described so far can also be found in PCT/EP00/00827. Inorder that the photoablation profile so elaborately computed canactually be implemented, the computer 48 is now programmed according tothe present invention such that the effect of the angle between thelaser beam and the corneal surface on the ablation depth is taken intoaccount.

As already mentioned, there are two significant factors involved inthis:

-   -   1) the laser-beam spot changes its size and shape as a function        of angle when it is incident on a curved surface, so that the        energy density of the incident laser beam changes, and    -   2) depending on the angle between the laser beam and the corneal        surface, a different fraction of the incident laser energy is        reflected away.

The effective, i.e. ablating energy density is hence reduced as afunction of the angle between the laser beam and the corneal surface.

Dependency of the Ablation Depth on the Effective Energy Density

It is therefore necessary first of all to examine the way in which thedifferent effective energy density affects the ablation depth.

This is represented in FIG. 4. The square points in this case stand forvalues measured with laser pulses of a particular length (for laserradiation from an ArF excimer laser with a wavelength of 193 nm). It canbe seen that the ablation depth increases with the logarithm of theeffective beam energy density. The ablation depth d therefore obeys theformula $\begin{matrix}{{d = {m \cdot {\ln\left( \frac{F}{F_{th}} \right)}}},} & (1)\end{matrix}$where F is the effective beam energy density and F_(th) is anenergy-density threshold above which ablation actually starts to takeplace. The factor m is a constant. The curve 52 was fitted according tothis formula. The energy-density threshold F_(th) was found to be 50mJ/cm² in this case.

The fact that the ablation depth as a function of the effective beamenergy density obeys such a simple formula facilitates numericalprocessing in the computer 48.

Incident Energy Density when Incident on the Curved Surface

The way in which the incident energy density changes as a function ofthe location of the incidence of the laser-beam spot on the cornea willnow be examined below.

FIG. 5 schematically shows the cornea 54, assumed to be spherical, onwhich a laser beam 56 is incident. Here, for simplicity, it is initiallyassumed that the laser beam 56 propagates parallel to the z axis. Thelaser-beam spot has an area A_(eff) on the surface of the cornea 54.

The area A_(eff) can now be calculated as a function of the coordinatesof the incidence point of the laser-beam spot centre on the cornea 54.

In this case, only two of the coordinates are independent of each other,the third coordinate being given by the shape of the surface of thecornea 54. The following hence applies for the z coordinate as afunction of the coordinates x and y:z=f(x,y)=f(r)={square root}{square root over (R ² −x ¹ −y ¹)}={squareroot}{square root over (R ² −r ²)}.   (2)

Here, R is the radius of the corneal hemisphere. r={square root}{squareroot over ((X²+Y² )} is the distance from the z axis to the incidencepoint 58 of the laser-beam spot centre.

If r₃ is the radius of the laser-beam spot before incidence on thecornea, then the following is obtained for A_(eff)(r): $\begin{matrix}{{A_{eff}(r)} = {\int_{- {rz}}^{rz}{\int_{{- \sqrt{{rz}^{2} - z^{2}}} + r}^{\sqrt{{rz}^{2} - z^{2}} + r}{\sqrt{1 + \left( {\frac{\mathbb{d}}{\mathbb{d}x}{f\left( {x,y} \right)}} \right)^{2} + \left( {\frac{\mathbb{d}}{\mathbb{d}y}{f\left( {x,y} \right)}} \right)^{2}}{\mathbb{d}x}{{\mathbb{d}y}.}}}}} & (3)\end{matrix}$

(This formula is found in Chapter F of: Höhere Mathematik griffbereit[higher mathematics ready-reference], Akademie-Verlag, Berlin, 1972,pages 638 to 643.)

The area A_(eff) of the cornea 54 is therefore greater by a factork1(r), $\begin{matrix}{{{{k1}(r)} = {\frac{A_{eff}(r)}{A_{0}} = \frac{A_{eff}(r)}{\pi \cdot r_{z}^{2}}}},} & (4)\end{matrix}$than the area A₀ of a laser-beam spot with normal incidence.

The beam energy density, moreover, is in fact defined as the ratio ofthe pulse energy of the laser E to the exposed A area, F=E/A. Thedensity of the energy incident on the corneal surface is thereforereduced to the value F/k1(r) in relation to the energy density F of theincident laser-beam spot.

Thus, with the known logarithmic dependency of the ablation depth on theeffective beam energy density, it is possible to set up a correctionfactor kor1(r), by which the ablation depth achieved with normalincidence of the laser-beam spot needs to be multiplied in order toobtain the ablation depth that is achieved in the case as represented inFIG. 5. This first correction factor is given as $\begin{matrix}{{{kor1}(r)} = {\frac{\ln\left( \frac{F}{{{k1}(r)}\quad F_{th}} \right)}{\ln\left( \frac{F}{F_{th}} \right)}.}} & (5)\end{matrix}$

FIG. 6 shows numerically determined curves in this context for differentradii R of the cornea 54.

As can be seen from FIG. 6, a significant deviation of the ablationdepth from the value 1, which was assumed in the prior-art methods, isfound towards the edge of the cornea.

The computer 48 is programmed in such a way that it compensates for thisreduced ablation depth, that is to say, for example, a correspondinglygreater number of laser-beam pulses are sent onto the relevant locationsso that the desired photoablation profile will be achieved.

Effect of Surface Reflection

We will next consider the dependency of the ablation depth on thesurface reflection. In FIG. 7, the incidence angle α₁ between theincident laser beam 60 and the normal 62 to the surface of the cornea isdefined, the cornea being represented here schematically in section as asemicircle 64.

To identify the reflected fraction of the incident light, use is made ofthe Fresnel equations, which can be found for example in the Lehrbuchder Experimentalphysik [textbook of experimental physics] by Bergmann,Schaefer, Volume III Optics, Walter de Gruyter, Berlin, N.Y. 1987, page496: $\begin{matrix}{{q_{\bot}\left( \alpha_{l} \right)} = \frac{\sqrt{n^{2} - {\sin^{2}\left( \alpha_{l} \right)}} - {\cos\left( \alpha_{l} \right)}}{l - n^{2}}} & (6) \\{{{q_{\parallel}\left( \alpha_{l} \right)} = \frac{{n^{2}\quad{\cos\left( \alpha_{l} \right)}} - \sqrt{n^{2} - {\sin^{2}\left( \alpha_{l} \right)}}}{{n^{2}\quad{\cos\left( \alpha_{l} \right)}} + \sqrt{n^{2} - {\sin^{2}\left( \alpha_{l} \right)}}}},} & (7)\end{matrix}$where q⊥ stands for perpendicularly polarised light and q∥ stands forparallel-polarised light and n is the refractive index of the cornealmaterial, which is for example n=1.52 for a wavelength of 193 nm (see G.H. Pettit, M. N. Ediger, Corneal-tissue absorption coefficients for 193-and 213-nm ultraviolet radiation, Appl. Optics 1996, volume 35, pages3386 to 3391). In order to obtain a dependency on the distance r, thefollowing formula is used $\begin{matrix}\begin{matrix}{{{\alpha_{l}(r)} = {{atan}\left( \frac{r}{\sqrt{R^{2} - r^{2}}} \right)}},} & {{where}\text{:}} & {0 \leq r^{2} < {{R\quad\lbrack{sic}\rbrack}.}}\end{matrix} & (8)\end{matrix}$

For unpolarised light, the reflectance k2(r) at the interface betweenair and tissue is given as:${{k2}(r)} = {\frac{{q_{\bot}^{2}(r)} + {q_{\parallel}^{2}(r)}}{2}.}$

If only the fact that a part of the incident radiation is reflected awaywere to be taken into account, and the aforementioned reduction in theenergy density due to the increase in the effective area A_(eff) inrelation to the original area A₀ were to be omitted, then an effectivebeam energy density of (1−k2(r))×F would be obtained in relation to theincident beam energy density F, and hence an attenuation of the ablationdepth d to kor2(r)×d, where $\begin{matrix}{{{kor2}(r)} = {\frac{\ln\left( \frac{\left( {1 - {{k2}(r)}} \right) \cdot F}{F_{th}} \right)}{\ln\left( \frac{F}{F_{th}} \right)}.}} & (9)\end{matrix}$

FIG. 8 represents numerically determined curves which represented [sic]the variation of kor2 as a function of the distance r of the incidencepoint of the laser-beam spot centre on the cornea from the z axis forvarious radii R of the cornea 64. As can be seen, the drop in theablation depth is very pronounced, especially at the edge, so that thedefects apparently were particularly large in the prior art, where avalue of 1 for kor2(r) was assumed even at the edge.

Combination of the Said Effects

Naturally, the two aforementioned phenomena, namely the increase in theeffective area of the laser-beam spot and the reflection, ought to betaken into account in combination with each other:

Of the incident beam energy density F, a beam energy density F/k1(r) isincident on the corneal surface, and the fraction (1−k2(r))×F/k1(r) ofthis is not reflected away.

A combined correction factor kor(r) for the ablation depth is hencegiven as: $\begin{matrix}{{{kor}(r)} = {\frac{\ln\left( \frac{\left( {1 - {{k2}(r)}} \right) \cdot F}{{{k1}(r)} \cdot F_{th}} \right)}{\ln\left( \frac{F}{F_{th}} \right)}.}} & (10)\end{matrix}$

The variation of this combined correction factor kor(r) is representedin FIG. 9 with the aid of numerically determined values as a function ofthe distance r of the incidence point of the laser-beam spot centre onthe cornea from the z axis for different radii of curvature of thecornea with a beam energy density of 150 mJ/cm².

As can be seen in FIG. 9, the drop in the correction factor kor(r)towards the edge is commensurately steeper where the radius of curvatureof the corneal surface is smaller.

This can also be seen with the aid of FIG. 10. There, the distancer_(80%), at which kor(r_(80%))=0.8, is divided by the value r_(max), atwhich kor(r_(max))=0, and this ratio is plotted against the energydensity of the incident laser beam.

With the approximation that the laser-beam spot is projected parallel tothe z axis, it is already possible to obtain satisfactory results. Ifthe laser beam is controlled by using the control program in thecomputer 48 so as to compensate for the reduction in the ablation depthat the edge of the cornea, more satisfactory healing results can beachieved for the patient's treatment.

Oblique Incidence of the Laser Beam

In a real system, however, it is rather the situation shown in FIG. 11which is encountered. The head of the galvanometric scanner 32,represented in FIG. 3, which deflects the laser beam 34 onto the eye 10,is offset relative to the cornea 66, schematically shown here as asemicircle. A laser beam emerging perpendicularly from the scanner headwould be incident on the cornea at an offset distance r_(v) from the zaxis. It can furthermore be seen that the angle between the laser beam68 sent onto the cornea 66 and the normal 70 to the surface is increasedfrom α₁ to the angle α₁+α₂. This leads to stronger attenuation of theablation depth on the right-hand side in FIG. 11.

In order to calculate this ablation depth, it is merely necessary torotate the coordinate system so that the z axis again runs parallel tothe laser beam. The formulae indicated above can then once more beapplied.

The dependency of the correction factor kor(r) on the offset distancer_(v) is represented by numerical simulation in FIG. 12 for the cornealarrangement shown in FIG. 11. As the offset distance r_(v) increases,the correction factor kor(r) assumes an increasingly asymmetric shape.

A perfect system therefore takes into account not only the increase inthe effective area of the laser-beam spot and the reflection at thecorneal surface, but also the effect of the offset. This can besummarised by saying that the effect of the angle between the laser beamand the corneal surface is taken into account.

However, this angle is not always accurately known. The invention cantherefore be refined further.

Firstly, the approximation that the cornea is spherical is not generallycorrect. It is generally aspherical and also often has astigmatism. Thecornea therefore has differing radii of curvature at differentlocations. These radii of curvature can be measured with so-calledtopography systems. Using this information about the local radius ofcurvature, on the one hand it is then possible to calculate the anglebetween the laser beam and the corneal surface, the formula (8) beingused. The fraction of the laser-beam energy incident on the cornealsurface which is reflected away can thereby be derived from formulae (6)and (7). To calculate the effect of the angle between the laser beam andthe corneal surface on the energy density of the laser-beam spotincident on the corneal surface, it is furthermore possible to useformulae (2) and (3), in which case R is then only the local radius ofcurvature of the cornea.

For further refinement, the fact that the angle between the laser beamand the corneal surface also changes during the ablation should be takeninto account. If, for example, a sequence of 50 laser-beam spot pulseswere to be necessary according to a previous calculation in order toablate the cornea according to the ablation profile at a particularlocation, then it may be that the angle between the laser beam and thecorneal surface changes further, after each pulse and therefore eachpartial ablation, in such a way that a smaller fraction of the laserbeam is reflected away, for example, so that only 49 or 48 pulses areneeded instead of 50 pulses or, conversely, it may be that theconditions become less favourable during the ablation so that morepulses are needed instead of the 50 pulses originally calculated. Since,according to the formulae indicated above, the extent to which alaser-beam pulse ablates the cornea is known, this effect can already betaken into account a priori in a refined computation. For thiscalculation, the corneal surface may be simulated on the computer, orthe changing local radii of curvature of the cornea may be calculatedapproximately. The calculation would then be carried out just as in thecase discussed in the previous paragraph, with locally differing radiiof curvature being incorporated.

As already mentioned above in the description of FIG. 3, the movementsof the eye are followed during the ablation. Because of these movements,not only is it naturally necessary to track the ablation profile to beremoved, and to control the scanner 32 accordingly, but also the anglebetween'the laser beam and the corneal surface changes during this. Thischange is preferably also taken into account. The angle with respect tothe axis A can be calculated, and the angle between the laser beam andthe corneal surface can be derived therefrom.

The present invention has been described with reference to aspot-scanning system, although it can also be employed in full ablationas well as slit-scanning ablation.

Numerically standard approximations may be used for the formulaeemployed.

1. A method of laser corneal ablation comprising: generating alaser-beam profile having one or more elementary laser beams; projectingthe laser-beam profile onto a cornea and thereby ablating apredetermined ablation profile from the cornea; and adjusting thelaser-beam profile according to an effect of an angle between the one ormore elementary beams of the laser-beam profile and a surface of thecornea on an energy density of the laser-beam profile on the surface ofthe cornea.
 2. The method of claim 1 wherein adjusting the laser-beamprofile further comprises increasing a time interval that at least oneelementary beam is incident on the corneal surface at an incident pointof the corneal surface as a function of the distance of the incidentpoint of the elementary beam center on the cornea from an axis runningparallel to the elementary beam direction, which axis meets the cornealsurface at a substantially right angle.
 3. The method of claim 1 furthercomprising: adjusting the laser-beam profile based on the fact that theenergy density F of an emitted elementary beam of radius reduced toF/k1(r), in the case of a cornea assumed to be substantiallyhemispherical with radius R, when incident on its curved surface, where$\begin{matrix}{{{kl}(r)} = {\frac{A_{eff}(r)}{A_{0}} = \frac{A_{eff}(r)}{{\pi \cdot r_{2}}2}}} \\{and} \\{{A_{eff}(r)} = {\int_{- {rs}}^{rs}{\int\frac{\sqrt{{rs}^{2} - x^{2}} + r}{\sqrt{{rs}^{2} - x^{2}} + r}}}} \\{\quad{\sqrt{1 + \left( {\frac{\mathbb{d}\quad}{\mathbb{d}x}{f\left( {x,y} \right)}} \right)^{2} + \left( {\frac{\mathbb{d}\quad}{\mathbb{d}x}{f\left( {x,y} \right)}} \right)^{2}}{\mathbb{d}x}\quad{\mathbb{d}y}}} \\{with} \\{{z = {{f\left( {x,y} \right)} = {{f(r)} = {\sqrt{R^{2} - x^{2} - y^{2}} = \sqrt{R^{2} - r^{2}}}}}},} \\{{r = \left( {x^{2} + y^{2}} \right)^{L_{2}}},}\end{matrix}$ where x, y, z are the coordinates of the incidence pointof the elementary beam center in a Cartesian coordinate system in whichthe origin lies at the sphere centre of the cornea.
 4. The method ofclaim 1 further comprising adjusting the laser-beam profile based on adistance r of an incidence point of the center of the one or moreelementary beams on the cornea from an axis running parallel to the oneor more elementary beam directions which meets the corneal surface at asubstantially right angle (z axis).
 5. The method of claim 1 whereinadjusting the laser-beam profile includes adjusting the ablation depthof the one or more elementary beams based on the determination that theablation depth due to a particular elementary beam pulse is reduced to dkor(r) in relation to the ablation depth d in the case of normalincidence of the elementary beam when the elementary beam is incident onthe curved surface, where${{kor}(r)} = \frac{\ln\left( \frac{\left( {1 - {{k2}(r)}} \right) \cdot F}{k\quad{{l(r)} \cdot F_{th}}} \right)}{\ln\left( \frac{F}{F_{th}} \right)}$and F_(th) is the energy-density threshold above which ablation takesplace.
 6. The method of claim 1 further comprising adjusting the laserbeam profile based on the fraction of the one or more elementary beamsincident on the corneal surface that is reflected away.
 7. The method ofclaim 1 further comprising adjusting the laser-beam profile based on thefact that in the case of a cornea assumed to be substantially spherical,the unreflected fraction of the energy density F/k1(r) of the one ormore elementary beams incident on the curved surface is given as(1k2(r)·F/k1(r), where $\begin{matrix}{{{{k2}(r)} = \frac{{q^{\underset{\_}{2}}(r)} + {q^{\underset{\_}{2}}(r)}^{\prime}}{2}},} \\{with} \\{{{q'}\left( \alpha_{1} \right)} = \frac{\sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}} - {\cos\left( \alpha_{1} \right)}}{1 - n^{2}}} \\{{{q{''}}\quad\left( \alpha_{1} \right)} = \frac{{n^{2}{\cos\left( \alpha_{1} \right)}} - \sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}}}{{n^{2}{\cos\left( \alpha_{1} \right)}} + \sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}}}}\end{matrix}$ where n/2-α₁ is the angle between the one or moreelementary beams and the corneal surface, where${\alpha_{1}(r)} = {{a\quad{\tan\left( \frac{r}{\sqrt{R^{2} - r^{2}}} \right)}\quad{with}\quad 0} \leq r^{2} < {R^{2}.}}$and n is the empirically determined refractive index of the cornea atthe wavelength of the one or more elementary beams which are used. 8.The method of claim 1 further comprising: determining the local radiusof curvature of the cornea; and determining the angle between the one ormore elementary beams and the surface of the cornea based on thelocalradius of curvature.
 9. The method of claim 1 wherein adjusting thelaser-beam profile includes adjusting the laser-beam profile based onthe fact that the angle between the elementary beams and the cornealsurface changes during the ablation.
 10. The method of claim 1 furthercomprising: sensing movements of the eye during ablation; and adjustingthe laser-beam profile based on the effect of the eye movements on theenergy density of the laser-beam profile on the cornea.
 11. A method oflaser corneal ablation comprising: generating a laser-beam profilehaving one or more elementary laser beams; projecting the laser-beamprofile onto a and thereby ablating a predetermined ablation profilefrom the corneal; and adjusting the laser-beam profile according to aneffect of an angle between the one or more elementary beams and thecorneal surface on the fraction of laser-beam energy incident on thecorneal surface which is reflected away.
 12. The method of claim 11further comprising adjusting the laser beam profile based on a distancer of an incidence point of the center of the one or more elementarybeams on the cornea from an axis running parallel to the one or moreelementary beam directions which meets the corneal surface at asubstantially right angle (z axis).
 13. The method of claim 11 whereinadjusting the laser-beam profile further comprises: adjusting thelaser-beam profile based on the determination that, in the case of acornea assumed to be substantially hemispherical with radius R, theunreflected fraction of the energy density F of the one or moreelementary beam incident on the curved surface is given as (1-k2(r))·F,where $\begin{matrix}{{{{k2}(r)} = \frac{{{q'}^{2}(r)} + {{q{''}}^{2}(r)}}{2}},} \\{with} \\{{{q'}\left( \alpha_{1} \right)} = \frac{\sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}} - {\cos\left( \alpha_{1} \right)}}{1 - n^{2}}} \\{{{q{''}}\quad\left( \alpha_{1} \right)} = \frac{{n^{2}{\cos\left( \alpha_{1} \right)}} - \sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}}}{{n^{2}{\cos\left( \alpha_{1} \right)}} + \sqrt{n^{2} - {\sin^{2}\left( \alpha_{1} \right)}}}}\end{matrix}$ where n/2-α₁ is the angle between the one or moreelementary beams and the corneal surface, where${\alpha_{1}(r)} = {{a\quad{\tan\left( \frac{r}{\sqrt{R^{2} - r^{2}}} \right)}\quad{with}\quad 0} \leq r^{2} < {R^{2}.}}$and n is the empirically determined refractive index of the cornea atthe wavelength of the one or more elementary beams.
 14. The method ofclaim 11 wherein adjusting the laser-beam profile includes adjusting theablation depth of the one or more elementary beams based on thedetermination that the ablation depth due to a particular elementarybeam pulseis reduced to d kor(r) in relation to the ablation depth d inthe case of normal incidence of the elementary beam when the elementarybeam is incident on the curved surface, where${{kor}(r)} = \frac{\ln\left( \frac{\left( {1 - {{k2}(r)}} \right) \cdot F}{k\quad{{l(r)} \cdot F_{th}}} \right)}{\ln\left( \frac{F}{F_{th}} \right)}$and F_(th) is the energy-density threshold above which ablation takesplace.
 15. The method of claim 11 further comprising: determining thelocal radius of curvature of the cornea;and determining the anglebetween the one or more elementary beams and the surface of the corneabased on the local radius of curvature.
 16. The method of claim 11wherein adjusting the laser-beam profile includes adjusting thelaser-beam profile based on the fact that the angle between theelementary beams and the corneal surface changes during the ablation.17. The method of claim 11 further comprising: sensing movements of theeye during ablation; and adjusting the laser-beam profile based on theeffect of the eye movements on the energy density of the laser-beamprofile on the cornea.
 18. A device for photorefractive corneal surgeryof the eye comprising: an instrument for measuring the entire opticalsystem of the eye to be corrected; means for deriving an ablationprofile from the measured values; a laser-radiation source; and meansfor controlling the laser radiation source to ablate a cornea inaccordance with the ablation profile; wherein the means for controllingincludes means for adjusting the laser-radiation source according to aneffect of an angle between one or more elementary laser beams of thelaser-radiation source and a surface of the cornea on an energy densityof the laser-radiation source on the surface of the cornea; wherein themeans for controlling includes means for adjusting the laser-radiationsource according to an effect of an angle between one or more elementarylaser beams of the laser-radiation source and a surface of the cornea ona fraction of laser-radiation energy incident on the corneal surfacewhich is reflected away.
 19. The device of claim 18 wherein theinstrument for measuring the entire optical system of the eye to becorrected is adapted to determine the local radius of curvature of thecornea.
 20. The device of claim 18 further comprising means fordetermining the position of the eye during ablation and adjusting thelaser-radiation source in accordance therewith.